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ZanzabumX [31]
3 years ago
11

Step 3: write an equivalent expression for sin4x that does not contain powers of trigonometric functions greater than 1.

Mathematics
1 answer:
Len [333]3 years ago
4 0

Answer: I don't know if you wanted to write sin(4x) or sin^{4} (x) , but here we go:

ok, sin(4x) = sin(2x + 2x), and we know that:

sin (a + b) = sin(a)*cos(b) + sin(b)*cos(a)

then sin (2x + 2x) = sin(2x)*cos(2x) + cos(2x)*sin(2x) = 2cos(2x)*sin(2x)

So 2*cos(2x)*sin(2x) is equivalent of sin(4x)

If you writed sin^{4} (x) then:

sin^{4} (x) = sin^{2} (x)*sin^{2} (x)

and using that: sins^{2} (x) = \frac{1-cos(2x)}{2}

we have: sin^{2} (x)*sin^{2} (x) = \frac{(1-cos(2x))*(1-cos(2x))}{4} = \frac{1-2cos(2x) + cos^{2}(2x) }{4}

and using that: cos^{2} (x) = \frac{1 + cos(2x)}{2}

\frac{1-2cos(2x) + cos^{2}(2x) }{4} = \frac{1-2cos(2x) + \frac{1+cos(4x)}{2} }{4}

You can keep simplifying it, but there is your representation of sin^{4} (x)  that does not contain powers of trigonometric functions greater than 1.

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